A mixed-integer goal programming model for capital budgeting within a police department

0198-9715/81/040171-11502.00,0 ~ergamon Press Ltd

Cornpar EnI%“,,“. L’rhan S~\,rm., “0,. 6. pp. 17, to 18,. 1981 Prmted in Great Br~tam

A MIXED-INTEGER GOAL PROGRAMMING FOR CAPITAL BUDGETING WITHIN POLICE DEPARTMENT

MODEL A

BERNARDW. TAYLORand ARTHURJ. KEOWN College

of Business,

Virginia Polytechnic Institute and State University, Blacksburg. VA 24061, U.S.A.

Abstract-A difficult

problem for most police departments is allocating limited dollars for capital expenditures in the most efficient manner possible. Because of the varied objectives of police services as perceived by police administrators, government officials and citizens, traditional allocation criteria, such as cost-benefit analysis, are not viable. As such, this paper proposes the use of mixed-integer goal programming as a methodology for selecting police expenditure items given a limited budget and multiple, conflicting objectives. In order to demonstrate this technique, a case example involving 18 potential expenditure items for a moderately-sized police department will be employed. This example model will be solved using a predetermined priority structure, and the results interpreted. Several alternative priority structures that reflect the objectives of different interest groups will then be tested in order to demonstrate the flexibility of this approach.

INTRODUCTION ONE OF THE major responsibilities of a municipal government is to provide police services to its constituency. However, a major difficulty inherent in this responsibility is determining the most efficient allocation of the scarce resources available for police services. Naturally, residents of a community desire that their tax dollars be used in a manner that will result in the best police department possible. This allocation problem, known commonly as the capital budgeting problem, is typical of all forms of governments and governmental units. Stated simply, given a limited budget and a number of expenditure items, the problem is to select those items that will insure the maintenance and greatest improvement in services. In the past, the traditional quantitative method used by local governments for allocating budget dollars has been some form of cost-benefit analysis as suggested by Eckstein [l] and McKean [2], among others. However, in recent years it appears that there is a consensus of opinion that cost-benefit analysis has severe limitations as a decisionmaking criterion for government expenditures [3-51. Cost-benefit analysis of a set of expenditure items requires that benefits associated with conflicting community and governmental goals be expressed in some common measure, such as dollars. Unfortunately, the benefits associated with police services (as well as other government services such as fire protection, health and education) can rarely be measured in terms of a common value like dollars. These services must be measured in terms of increased efficiency in protecting human life and property, and enhancing the quality of life. In addition, there appears to be a consensus that decisions regarding budget expenditures must encompass trade-offs that reflect the conflicting goals between the government, the police department and the community [6-121. Thus, the traditional process of ranking budgetary items according to benefits measured in commensurable units in the costbenefit framework, is no longer viewed as an appropriate or efficient budgeting technique [4, 5; 133. As an alternative, this paper will demonstrate integer goal programming as an efficient and viable approach to the capital budgeting problem in police departments. THE GOAL PROGRAMMING METHODOLOGY A number of researchers including Stuart [14], Laidlaw [13] and Heroux and Wallace [15], have proposed the use of linear programming as an approach to the capital 171

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BERNARD W. TAYLOR and ARTHUR J. KEOWN

budgeting problem in public sector planning. However, while these efforts represent significant strides towards a solution to the capital budgeting problem, they still encompassed only a single objective function. They did not deal adequately with the problem of multiple conflicting goals and the measurement of these goals in incommensurable units [7, 91. Alternatively, Lee and Sevebeck [16], Awerbuch and Wallace [17], Ignizio [18] and several others [4, 19, 201 have recently suggested goal programming as a means of overcoming the difficulty of multiple conflicting goals in urban planning. Goal programming is a linear mathematical programming technique that allows for the consideration of multiple goals where those goals are prioritized and treated in a simultaneous and/or sequential manner. In a police department the existence of multiple conflicting goals is especially prevalent. For example, while the police department might seek to use budget dollars to improve its own physical facilities or modernize its record keeping system, the citizenry might desire to see the same budget dollars used to purchase more patrol cars in order to provide greater police visibility. In either case, the effectiveness (or benefits) of these expenditures cannot usually be measured in commensurable units. Goal programming avoids the traditional problems of translation of incommensurable goal measurements into common benefit or utility measurements by allowing goals to be measured in unlike units through the creation of a multidimensional objective function. This approach lends itself to decisions with incommensurable goals and allows deviations from these goals to be treated sequentially and/or simultaneously (see Charnes and Cooper [21], Hartley [22], Ignizio [23] and Lee [24]). However, although goal programming can deal effectively with the problem of multiple conflicting goals, in its general form it does not compensate for the indivisibility of inputs. In the capital budgeting problem in police departments, expenditure items are most often indivisible. For example, it is possible to purchase only whole patrol cars and not fractional portions. Thus, the decision variable associated with the purchase of patrol cars must be restricted to integer values. Similarly, some expenditures require either a yes or no decision (i.e. purchase or not purchase). For example, the establishment of a neighborhood crime prevention program would require a decision variable restricted to the integer values of zero or one. Moreover, some expenditure items are mutually exclusive and/or dependent on other items, and, also require integer solutions. However, when expenditure items are indivisible, Dantzig [3] and Weingartner [S] have shown that approaches other than mixed integer programming can produce suboptimal results. Thus, in order to compensate for multiple conflicting goals and the existence of indivisible inputs the most logical approach to the capital budgeting problem in police departments is integer goal programming. Since the topics of goal programming [23-25) and integer goal programming [26] have previously been described in detail and. integer goal programming has been applied to other urban capital bugeting problems, a general discussion of these topics will not be presented. Instead an illustrative example of the development of an integer goal programming model for a police department involved in the capital budgeting process will be described.

THE

GOAL

PROGRAMMING

MODEL

The purpose of presenting this case example is to demonstrate the applicability of integer goal programming to the capital bugeting activity of a law enforcement agency. The example involves a budgeting decision that encompasses 18 proposed expenditure items. Thirteen of these will have zero-one solution values while the remaining five will be integer. The items represent expenditures over and above the present operating budget for facilities and staff of a moderately-sized police department. Variables of the goal programming model define the purchase of a particular item or several items : xj = the selection

of an expenditure

item.

Capital budgeting within a police department

173

T,he proposed expenditures are as follows: x1 = x2 = x3 = x4 = x5 = x6 = x7 = xs = x9 = xl0 = xi 1 = xi2 = xi3 = x14 = xl5 = xl6 = xl7 = xi8 =

patrol cars (a maximum of 4) police retraining program community education program rape counseling and prevention program police dogs (a maximum of 2) non-lethal weapons (a maximum of 3) neighborhood youth council and juvenile education program station renovation juvenile holding area mobile traffic radar units (a maximum of 5) traffic violators education program hand-held transceivers (a maximum of 4) scrambler system mobile digital transmission and reception display system automated display record system electronic burglary system polygraph and renovated interrogation room mini-computer

Variables xl, x5, x4, xro and xl2 will have integer solutions limited by upper bounds as indicated above. Variables x2, xX, x4, x7, x8. x9, x1,, x13, xi4, xlSr xi& xl7 and .~,a will have zero-one solutions indicating that &he project is seIected or not selected. Many of the zero-one variables are for programs to be established and presented by the police department staff.

The goal constraints in this model can be categorized as follows. First there are the dollar budgeting constraints reflecting the initial expenditure for each item. and. the addition to the operating budget resulting from each expenditure. Next, goal constraints will be defined that reflect the improvement in police efficiency and service generated by the expenditure items. Finally, several constraints that reflect the conrin~]e~rrnature of some of the items will be developed. While other goal formulations would be justified. the purpose of this example is to demonstrate the robustness of the mixed integer goal programming approach to the law enforcement capital budgeting decision which is characterized by multiple goais measured in incommensurable units and both integer and non-integer solution variables. Budget constrainrs. The first goal constraints indicate the initial expenditure required for each item and the total amount available for expenditures (i.e. the capital budgeting goal). $7500X, + 4200x2 + 1200x, 4 1500x4 + 700x, f 250x6 + 800~~ + 6700.x8 + 3500.~~ +2100x,,

+ 5OOx1r + 375x,, + 6500xi3 -t- 5700x1, t 2500x,,

-I- 67OO,u,, + 6800x1,

+ 3400x18 + d; - d; = $47.500 In goal programming, d; and d: are deviational variables that reflect the amount by which the goal (i.e. $47,5~) is under-achieved (d;) or exceeded (d:). Since $47,500 is an upper budget limit, the goal is to avoid exceeding this amount. Thus, in the goal programming objective function we will attempt to minimize d:, the amount of excess spending over $47,500. However, since there will be other goals in competition (i.e. conflicting) with this goal, the goals are given priorities to signify their importance in relation to each other. Subsequently, the achievement of goals will be attempted in order of their priority. The remaining goals in this model are expressed similarly. <‘.f.~ S.6 4 8

BERNARD W. TAYLOR and ARTHUR J. KEOWN

174

The next goal constraint reflects the addition to the annual operating budget resulting from each of the 18 expenditure items. The goal level of $8000 in this constraint is the total increase in the annual operating budget. $800~1 + 1500x2 + 600x3 + 1000x4 + 900x5 + 50~~ + 600x,

+ 400~s + 200x9

+ 100x10 + 1500x11 + 45x12 + 175x13 + 550x14 + 700x15 + 300X16 +350x,,

+ 1800~~s + d;

- d:

= $8000

The goal for this constraint is to minimize positive deviation (di) as was the case for the first goal. EfJiciencj~ and sercice criteria. This next category of goal constraints reflects the desired improvement in police efficiency and service that will be achieved from each of the expenditure items. The goal levels, in general, represent a desired level of improvement over some base value presently being realized. These efficiency criterion goals can further be categorized into three areas of measurement: crime prevention potential, crime control, and, perceived quality of the law enforcement unit by citizens and peers. These various criteria have been identified as appropriate for measuring the effectiveness of law enforcement agencies [7, 8, 11, 12, 271. The first goal constraint represents the degree to which each expenditure item reduces the felony crime rate, measured by the number of felonies per 1000 inhabitants per year. As such, this is a crime prevention criterion. 1.3x, + 0.6~~ + 0.4x, +0.04x,2

+ 0.2x4 + 0.02~~ + 0.03x6 + 0.3x,

+ 0.08~~~ + 0.07x1‘, + 1.1~~~ + 0.03x1,

+ 0.02~~~ + 0.01x1,

+ 0.04~~~ + d;

- d;

= 7.0

In this goal constraint, the goal level of 7.0 is the total desired reduction in the felony crime rate from the existing felony crime rate. Each coefficient in the goal constraint represents the reduction in felonies per 1000 inhabitants per year for each expenditure item purchased. Notice that some of the expenditure items are not included in this constraint while others have an extremely small potential reduction. This is only natural since some items (such as station renovation or juvenile holding area) will not result in felony crime reduction and, others (such as the purchase of non-lethal weapons) will have only a negligible effect. The following constraint is another crime prevention goal for misdemeanor crime rute reduction. As in the preceding felony crime rate goal, the misdemeanor crime rate is measured by the number of misdemeanors per 1000 inhabitants per year. 3x, + 2x, + 0.6x3 + 0.5x5 + 1.3x, + 2xie + 1.6~~~ + 0.2~~~ + Ai - d4’ = 18.0 The goal level of 18.0 is the total desired reduction in the misdemeanor crime rate from the existing misdemeanor crime rate. The next constraint is also a crime prevention goal. It reflects the reduction in meruge proprrt_v losses per year resulting from all crimes. The constraint coefficients and goal level are measured in terms of an index computed by expressing the property loss per year as a percentage of the total property value in the community and multiplying this percentage by 100. The goal level of 17.0 is the desired reduction in the property loss index from the index that presently exists. 2x1 + 1.8~2 + 1.4x3 + 0.3~~ + 0.8x,

+ 0.1x,,,

+ 0.1x,, +2.1xi6

+ 4xi2 + 0.2~~~ + 0.5x1,

+ d;

- d;

= 17.0

The following two goals are criterion measures of crime control. The first goal constraint reflects the percentage of crimes cleared by arrest and assignment of guilt. The goal

Capital

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175

a police department

level of 10 is the desired increase in the percentage assignment of guilt.

of crimes cleared by arrest and

1x1 + 0.6x, + 0.02x, + 0.12x4 + 0.7x5 + 0.2x, + 0.07x, + 0.2~~~ + 0.4~~~ + 0.3~~~ + 0.2~~~ + 0.3~~~ + 0.6~~~ + 0.9x1, + d; - ds+ = 10 The next crime control goal constraint reflects the average time between the occurrence of a crime and apprehension. The goal level is a desired reduction of 1 day in the present average time required between the crime and arrest. 0.03x1 + 0.04x2 + 0.1x4 + 0.2x, + 0.01x1* + 0.02x13 + 0.2x14 + 0.2x15 + 0.08~~~ + 1.3x1, + 0.08~~~ + d; - d: = 1 The following three goal constraints are measures of the perceived quality and effectiveness of the police department by the citizenry and by outside peer groups. The first goal constraint is a measure of the security felt by the citizens. This measurement is expressed as an index from 0 (no feeling of security) to 100 (totally secure) and is developed from a public opinion poll. The goal level represents the desired increase in this index above the present level. 0.4~~ + 0.1~~ + 0.6x3 + 0.7x4 + 0.1x5 + 0.3x, + 0.1~s + 0.1~9 + 0.08~11 +o.07x15 + 0.3~~~ + 0.05x1, + d, - d,+ = 2.5 The second goal constraint reflecting efficiency, service and quality is for positive citizen contacts. This criterion is expressed as the percentage of total citizen contacts with the police department that are viewed by the citizenry as being positive (i.e. the citizen had a favorable impression as opposed to negative or unfavorable). As in the previous goal constraint, this measure is also determined by a public opinion poll. The poll level of 0.5 represents the desired increased percentage of positive citizen contacts. 0.2x, + 0.4~~ + 0.9x3 + 0.8x4 + 0.02x5 + 0.03x6 + 0.7x, + 1.1~~ + 0.7x9 + 0.2~~~ + 0.4~~~ + 0.15~~~ + 0.4~~~ +d,

- d,+ = 5.0

The final goal constraint reflecting departmental efficiency, service and quality is for an external quality rating factor. This rating is an index (with a perfect score being 100) that is determined by an outside rating team consisting of administrators and officers from other communities and officials of police organizations. The rating system emphasizes the quality of facilities and service. The goal level in this case represents the desired increase in the present rating for the police department. 0.1~~ + 0.6x2 + 0.3x3 + 0.2x4 + 0.06~~ + 0.4~~ + 0.5x, + 0.8x; + 0.7x9 +0.07x10 + 0.15x11 + 0.2x12 + 0.3x13 + 05x14 + 0.9x15 + 0.3X16 + 0.7x1, + 0.7~~~ + d,

- d:,

= 6.5

Mutually exclusive and contingent items. These goal constraints represent categories of items that are contingent upon the selection of other items for selection. For example, the following goal constraint reflects the fact that the mobile digital transmission and reception display system (xr4) and the automated display record system (xIs) are contingent upon the purchase of the mini-computer (xra). The goal constraint is expressed as, x14

+

xl5

-

2x1* + d;, - d:, = 0.0

Next, the renovation of the juvenile holding area (x9) and the purchase of the polygraph and renovated interrogation room (x1 ,) are dependent on station renovation (xa): x9

+

x17

-

2xs + d,

- d12 = 0.0

176

BERNARD W. TAYLOR and ARTHUR

Due to limited contact with the public at least one of the four education programs x3 +

x4 +

x7 +

However, due to limited staffing cation programs can be initiated:

J.

KEOWN

at the present time, it has been should be instigated:

x11 +

d13 -

capabilities

determined

dc3 = 1.0

a maximum

of three of the four edu-

~~+~~+x~+xii+d;~-d:~=3.0 If two or more new patrol cars are purchased traffic radar unit will be required: Xl

100x10 + d;,

-

If three or more mobile traffic radar education program must be instigated: x10 -

units

then at least one additional

mobile

- d:5 = 1.0 are purchased

then the traffic violators

1OOxii + dT6 - d& = 2.0

These last two constraints can be interpreted as follows. In the first constraint, d15 will be minimized in the objective function, thus, inhibiting the goal constraint from achieving a value greater than 1.0. Thus, if zero or one patrol car (xi) is purchased the constraint will not be violated. However, if two patrol cars are purchased then at least one radar unit (xio) must be purchased in order to achieve the goal of minimizing dc6. This will result in a large negative value which will indeed achieve the stated goal. .Regardless of the number of patrol cars that are purchased (if any) the number of radar units (xi i) is not restricted. The same logic is true for the second constraint. Maximum purchase constraints. As indicated in the definitions of the model variables, some of the variables have integer solutions with upper limits, as follows: x1 x5 x.5 xl0 xl2

+ + + + +

d;, dF8 d, d, d;,

-

dc7 d18 d:, dlo d:,

= = = = =

4 2 3 5 4

(patrol cars) (police dogs) (non-lethal weapons) (mobile traffic radar units) (hand-held transceivers)

All other decision variables involve zero-one decisions. The minimization of positive deviation in these constraints at the highest priority level makes these strict constraints, in which the upper limit cannot be exceeded. THE

The general

goal programming

OBJECTIVE

objective min

FUNCTION

function

f

i

j=l

k=l

is formulated

as follows:

PjWkd,

where Wk = inter-priority weighting factors (k = 1,. . . n); dk = deviational variables in Pj = preemptive (lexiographic) priority factor which minimization is desired; (j = 1,. . m), such that no number n, however large, can make nPj+ I greater than or equal t0 Pj. The objective function involves the minimization of deviational variables, which are weighted and/or ranked, from desired goal levels. Thus, when the deviational variable from a specific goal is driven to zero, that particular goal has been reached and is fully satisfied. As such, the deviational variables make up the heart of the objective function. However, goal programming allows these variables to be treated in a sequential manner through the use of preemptive priorities in addition to allowing for the weighting of deviational variables when more than one deviational variable is included at a priority level. Thus, the use of goal programming allows for a multidimensional objective function in which ordering and weighting of goals is employed to deal with the problem of incompatible multiple goals.

Capital

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177

Incompatible or incommensurable goals measured in unlike units are ranked so that goals given a lower rank are satisfied only after those given higher ranking are totally satisfied or have reached a point beyond which improvement is impossible. This is done through the use of preemptive (lexiographic) priority factors Pj(j = 1,. . m). The preemptive priority factors are interpreted such that Pj B Pi+ I implies that no number n, however large, can make nPj+ 1 g reater than or equal to Pi. If several goals are deemed compatible they can be treated at the same priority level through the use of weights, IV, (k = 1,. . m). The criterion used to determine the weight attached to each deviation should be to weight each variable with a factor representing the relative amount of ‘disutility’ associated with one unit of unsatisfactory derivation from the goal level. Priority structure The priority structure of goals in a public organization is typically determined by a group of individuals. For our case example, this would include police department administrators, local government officials and possibly citizen representatives. This group has established the following goals in order of their importance. Strict constraints

= Those goals related limits. The positive all of these.

to contingent purchases and strict upper deviational variables are minimized for

P, = The most important administrative expenditures to a limit of $47,500.

priority

is to limit capital

P2 = The second priority is to limit any increase in annual operating expenses resulting from the purchase of items to a level of $8000 per year. P3 = The third priority is to achieve the felony crime rate reduction goal of 7 felonies per 1000 inhabitants per year. P4 = The fourth priority is to achieve the misdemeanor crime rate reduction goal of 18 misdemeanors per 1000 inhabitants per year. P5 = The fifth priority is to achieve the two goals for crime control (i.e. percentage of crimes cleared by arrest and the average time between occurrence of a crime and apprehension). P6 = The sixth priority

This priority

is to achieve

the property

loss goal.

P, = The seventh priority is to achieve the goals for citizen ity and positive contacts.

secur-

P8 = The final priority factor goal of 20.

rating

structure

results

in the following

is to achieve objective

the external

quality

function:

minimize Z = PO[

y

d:

+ d;,]

+ P,d:

+ Pzd:

+ P3d;

+ P4d;

i=ll if13 +

P,(d,

+ d;)

+ PJ;

MODEL

+ P,(d,

+ d,)

+ PsdcO

RESULTS

The integer goal programming model involved 18 variables, 11 strict constraints, 10 goal constraints, and an objective function. The model was solved by the Lee and Morris algorithm [26] which is based upon an extension of the branch and bound algorithm of Land and Doig [28] as modified by Dakin [29].

BERNARDW. TAYLOR and ARTHUR J. KEOWN

178

The solution to this problem resulted in the recommended expenditure items: x1 x2 x5 x6 x, xl0 xlZ

= = = = = = =

x16

=

purchase of the following

4.0 (patrol cars) 1.0 (retraining program) 2.0 (police dogs) 3.0 (non-lethal weapons) 1.0 (youth council and educational programs) 1.0 (mobile traffic radar unit) 4.0 (hand-held transceivers) 1.0 (electronic burglary system)

The model completely satisfied goals P, through P, (for strict constraints, capital expenditure limitation, annual operating expenses, felony crime rate and misdemeanor crime rate). However, goals P, through PB were not satisfied. The degree of underachievement of these goals was as follows: Priority 5 = Underachieved. The desired 10% increase in the percent of crimes cleared by arrest fell short by 0.93% and the desired 1 day decrease in the average time between occurrence of a crime and apprehension fell short by 0.32 days. Priority 6 = Underachieved. The desired reduction in the property loss index of 17 points, fell short by 2.0 points. Priority 7 = Underachieved. While the desired increase in the public security index of 2.5 points was exactly met, the desired increase of 5% of total citizen contacts with the police department fell short by 2.82%. Priority 8 = Underachieved. The desired increase in the external quality rating index of 6.5 points fell short by 2.51 points. SENSITIVITY

ANALYSIS

While the above solution was based upon the priority structure resulting from the municipalities preferences and desires with respect to the selection of new law enforcement items, it is by no means the only possible formulation. In fact this methodology can be used to identify resource requirements necessary to attain all the desired goals or goal attainments under various combinations of inputs and priority structures of goals. As such, this analysis could be extended to a sensitivity analysis of the optimal solution through the use of discrete changes in the priority structure of goals, goal levels and technological coefficients. One such application of sensitivity analysis would involve a restructuring of the objective function priority structure to reflect a greater sensitivity to citizen and external perception of the police department. This would be accomplished by moving the citizen security and positive contacts goals (formerly at the seventh priority level) and the external quality rating factor goal (formerly at the eighth priority level) to the first and second priority levels respectively. This would result in the following objective function: minimize z = p0

z [

+

1

&+ + dam + P,(d;

i=ll i#13

P4d;

+ P5d;

+ P6d;

+ d;) + Pzdlo + Psd:

+ P,(d;

+ d;) + P,d;.

The minimization of this objective function results in the following variable selection : x1 = 2.0 (patrol cars)

x3 = 1.0 (community education program) xq = 1.0 (rape program)

Capital

x6 x, x8 x9 xl0 xl2 xl5 x16 xl8

= = = = = = = = =

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179

3.0 (non-lethal weapons) 1.0 (youth council and juvenile program) 1.0 (station renovation) 1.0 (juvenile holding area) 1.0 (mobile radar unit) 4.0 (hand-held transceivers) 1.0 (automated record system) 1.0 (electronic burglary system) 1.0 (mini-computer)

This variation of the model completely satisfied the first five goals (PO through P4); however, it failed to satisfy goals P5 through P8. The degree of underachievement on these goals was as follows: Priority 5 = Underachieved. The desired reduction in the felony crime rate measured by the number of felonies per 1000 inhabitants per year of 7.0 points fell short by 2.09 points. Priority 6 = Underachieved. The desired reduction in the misdemeanor crime rate of 18 fell short by 8.10. Priority 7 = Underachieved. The desired reduction in the percent of crimes cleared by arrest and assignment of guilt of 10% was underachieved by 4.49%, while the desired reduction in the average time between the occurrence of a crime and apprehension of 1 day fell short by 0.44 days. Priority 8 = Underachieved. The desired reduction in the property loss index of 17 points from the acceptance of new projects fell short by 7.0 points. The results and goal achievements which reflect a greater sensitivity to citizen and external perception of the police department could then be compared to the results and goal achievements from the original model formulation, In this way an evaluation of the budgeting decision problem facing the police department can logically evolve. At this point further sensitivity analysis can be performed to determine what level of funding would be required to achieve all the desired goals. This can be accomplished by moving the two budget constraints to the seventh and eighth priority levels. This would result in the following objective function: minimize + Pld; [ jldi++d;,]

z=f’o

+ Pzd;

+ P,(d,

+ d; ) + P4d;

if13

+

P,(d,

+

d;)

+ P6d;O + P,d:

+ Psd:.

The minimization of this objective function results in the following variable selection : x1 = 4.0 (patrol cars)

x2 = x3 = x5 = x6 = x, = x8 = x9 = xl0 = x1 1 = xl2 =

1.0 (retraining program) 1.0 (community education program) 2.0 (police dogs) 3.0 (non-lethal weapons) 1.0 (youth council and juvenile program) 1.0 (station renovation) 1.0 (juvenile holding area) 2.0 (mobile traffic radar units) 1.0 (traffic violators education program) 4.0 (hand-held transceivers)

BERNARD W. TAYLOR and ARTHUR J. KEOWN

180 xl6 = 1.0 (electronic xl7 = 1.0 (polygraph

burglary system) and renovated interrogation

room)

While this variation of the model completely satisfied the first seven goals (PO through P6), it failed to satisfy the two budget goals P, and Ps. The degree of underachievement on these goals was as follows: Priority

7 = Underachieved. The desire to limit the level of total initial expenditures to $47,500 was exceeded by $20,750 as total expenditures necessary to achieve the first seven priorities (PO through P6) was $68.250.

Priority

8 = Underachieved. The desired $8000 limit on the increase in the annual operating budget as a result of new expenditure items was exceeded by $2980. The increase in the annual operating budget because of the acceptance of all expenditure items was $10,980.

Through the use of sensitivity analysis the decision maker is given the opportunity to not only determine the level of expenditures necessary to satisfy all goals, but, also to examine the effect of alternating the goal priority structure on the project selection process. SUMMARY

It has been the purpose of this paper to demonstrate a more effective means for evaluating annual capital budgeting expenditures for a police department. Integer goal programming was shown, via a case example, to be a viable approach for determining which capital expenditure items should be selected in an environment where multiple and conflicting goals exist. In addition the capabilities of this technique for peforming sensitivity analysis related to alternative goal priority structures was demonstrated. REFERENCES The Economics of A Project Ecaluarion, Harvard Univ. Press, 1. Eckstein 0. Water Resource Development: New York (1958). in Government Through Systems Anal&, Wiley, New York (1958). 2. McKean R. N. Eficiencg problems with some integer variables. 3. Dantzig G. B. On the significance of solving linear programming Econometrica 28, 3W4 (1960). approach. 4. Keown A. J. and Martin J. D. Capital budgeting in the public sector: a zero-one programming Financial Mgf 7, 21-27 (1978). H. M. kfAfhemAfiCA/ Programming And the Ana/gsis of Capitul Budgeting Problems, Prentice5. Weingartner Hall, Englewood Cliffs, NJ (1963). M. Social structures for the enhancement of scientific information in urban 6. Bozeman B. and Fitzgerald policy-making and management, Urban Systems 3. 163-175 (1979). in planning state and local programs, Decision-Making in Urban 1. Hatry H. P. Criteria for evaluation Planning, I. M. Robinson (Ed.), pp. 209-240, Sage, Beverly Hills, NJ (1972). measurement, Readings on Productiaity in 8. Hatry H. P. Wrestling with police crime control productivity Policy, J. L. Wolfe and J. F. Heaphy (Eds), pp. 86128, Heath (Lexington Books), Lexington, MA (1975). J. L. and Unsinger P. C. Community Police Administration, Nelson-Hall, Chicago (1975). 9. Kuykendall in Urban Planning, pp. 21-32, Sage. Beverly Hills, CA (1972). 10. Robinson I. M. Decision-hfaking P. M. and Ferguson R. F. The Managing of Police Orgunizations (2nd edn), Prentice-Hall, 11. Whisenand Englewood Cliffs, NJ (1978). (4th edn), McGraw-Hill, New York (1977). 12. Wilson 0. W. and McLaren R. C. Police Administration 13. Laidlaw C. 0. Linear Programming for (irban Development Plan Eraluution, Praeger, New York (1972). programming models. Socio-Econ. Plan. Sci. 217-237 (1970). 14. Stuart D. G. Urban improvement 15. Heroux R. L. and Wallace W. A. Financial Ancrlysis and the New Community Decelopment Process, Praeger, New York (1975). model for municipal economic planning. Policy Sci. 99-115 16. Lee S. M. and Sevebeck W. R., An aggregative (1971). 17. Awerbuch S. and Wallace W. A. Policy Ecahation /i)r Community Development. Decision Tools for Local Government, Praeger, New York (1976). to goal programming with applications in urban systems. Urban Sysfems 5, 18. Ignizio J. P. An introduction 15-33 (1980). 19. Lee S. M. and Keown A. J. Integer goal programming for urban renewal planning. Urban Systems 4, 17-26 (1979). urban recreational facilities with integer goal programming. J. 20. Taylor B. W. and Keown A. J. Planning Opls. Res. Sot. 29. 751-758 (1978).

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